Arithmetic I INTRODUCTION Arithmetic, branch of mathematics that arises from counting, the most basic mathematical operation. Arithmetic encompasses various ways of counting, or manipulating numbers: addition, subtraction, multiplication, and division. The ancient Greek word arithm?tik? combined the words arithmos, meaning "number," and techn?, referring to an art or skill. Thus, arithmetic means the art of numbers. The numbers used in arithmetic may be positive integers (whole numbers), negative integers, fractions, or decimals. For the history of arithmetic and mathematics, see Mathematics. Different civilizations throughout history have developed different kinds of number systems. Although the ancient Babylonians used a system based on the number 60, all modern cultures employ a system in which objects are counted in groups of ten, probably because humans have ten fingers and tend to use them in counting. This system is called the decimal, or base 10, system. II OPERATIONS WITH POSITIVE INTEGERS We generally count with positive integers. This unending sequence of whole numbers starts with 1. Each subsequent number in the sequence is one more than the number before: 1, 2, 3, 4 .... Alternating numbers starting with 1 (1, 3, 5 ...) are called odd numbers, while every other number starting with 2 (2, 4, 6 ...) is called an even number. In numbers with two or more digits, every digit has a so-called place value. In the decimal system, the place value increases from units, or ones, to tens, hundreds, thousands, and higher as the number of digits increases from right to left. We can also say that each place increases by a power of 10. A power of a number is the number of times the number is multiplied by itself. In the number 1,111, for example, the place value on the far right is a unit or ones value; the place value just to its left is 10 (1 × 10); the next one to the left is 10 × 10, or 100; and the place value on the far left is 10 × 10 × 10, or 1,000. The number 2,534, then, is equivalent to (2 × 1,000) + (5 × 100) + (3 × 10) + (4 × 1). A Adding Positive Integers The arithmetic operation of addition is basically a means of counting quickly and is indicated by the plus sign (+). We could place 4 apples and 5 more apples in a row, then count them individually from 1 to 9. Addition, however, makes it possible to count all of the apples in a single step (4 + 5 = 9). We call the end result of addition the sum. The simplest sums are usually memorized. This table shows the sums of any two numbers between zero and nine: To find the sum of any two numbers from 0 to 9, locate one of the numbers in the vertical column on the left side of the table and the other number in the horizontal row at the top. The sum is the number in the body of the table that lies at the intersection of the column and row that have been selected. For example, 6 + 7 = 13. We can easily add long lists of numbers with more than one digit by repeatedly adding one digit at a time. For example, if the numbers 27, 32, and 49 are listed in a column so that all the units are in a line, all the tens are in a line, and so on, finding their sum is relatively simple: First add the units (7 + 2 + 9); they total 18. Then add the digits in the tens place (2 + 3 + 4); they total 9, but this means 9 tens, or 90. In the last step, add the total of the units to the total of the tens: We can skip the second step, adding the sum of the units to the sum of the tens, by using a shortcut called carrying. Carry the 1 in 18, which stands for 1 ten, over to the tens column and add it directly to the digits there: Add the digits in the tens column, including the carried 1, and place the sum, 10, just to left of the units sum. The result is 108. Similarly, when adding numbers with three or more places, we can carry digits to the hundreds place, thousands place, or beyond. B Subtracting Positive Integers The arithmetic operation of subtraction is the opposite of addition and is indicated by the minus sign (-). If we take 5 apples away from 9 apples, subtraction tells how many apples remain without our actually counting them. The simple sums memorized for addition are used in reverse for subtraction. For example, the result of 9 minus 5 is 4 because 4 is the number we would have to add to 5 for a sum of 9. The end result of subtraction is called the difference. It is possible to subtract 23 from 66 by counting backward 23 integers from 66, one number at a time, or by taking away 23 items from a collection of 66 and counting the remainder. Either way we would reach 43. The rules of arithmetic for subtraction, however, provide a much quicker method for obtaining the answer. We can subtract large numbers by repeatedly subtracting one digit at a time. First align the numbers under one another, units under units, tens under tens, as in addition: Subtract the units: 6 - 3 = 3. Then subtract the tens column: 6 - 2 = 4. The results of these two single-digit subtractions, written side by side, provide the answer: Subtraction is a bit more complicated if we need to subtract a larger digit from a smaller one. For example, when subtracting 47 from 92, the units value (7) of 47 is greater than the units value (2) of 92. We can handle this situation using a procedure called borrowing, which is like carrying in reverse. Ten units can be borrowed from the tens column--that is, from the 9 of 92--leaving 8 in the tens column. Bring the 10 over to the units column and add it to the 2 already there, giving 12 in that column from which 7 can then be subtracted: Complete the subtraction by taking 4 away from 8 in the tens column, which gives 4. The answer, or difference, is 45. C Multiplying Positive Integers Multiplication is simply repeated addition and is often indicated by the times sign (×). The expression 3 × 4 means that 3 is to be added to itself 4 times or, similarly, that 4 is to be added to itself 3 times. In either case, the answer is the same: 12. For example, 3 sets of 4 apples together contain a total of 12 apples. When large numbers are involved, however, such repeated addition is tedious. Multiplication provides a procedure for simplifying repeated addition. Sometimes a dot or an asterisk is used instead of a times sign to indicate the multiplication of two or more numbers, and sometimes parentheses are used. For example, 3 × 4, 3 · 4, 3 * 4, and (3)(4) all indicate 3 times 4. The end result of multiplication is called the product. As in addition, it is customary to memorize the products of the integers between 0 and 9, which are provided in the following table: To find the product of any two numbers from 0 to 9, locate one of the numbers in the vertical column on the left side of the table and the other number in the horizontal row at the top. The product is the number in the body of the table that lies at the intersection of the column and row that have been selected. For example, 7 × 8 = 56. In order to multiply numbers with more than one digit, align the units, tens, hundreds, and so on. The order in which two numbers are multiplied does not affect their product: 3 × 4 = 12 and 4 × 3 = 12. It is generally easier, however, to multiply two numbers if the number with the most digits is placed on the top: We then multiply each digit of the top number by the bottom number, in this case, 4. Adding the results of all these multiplications together gives the product 1,544: Carrying tens and hundreds, as in addition, shortens this operation: Multiply the 6 by 4, giving 24. Write the 4 in the units place of the product and carry the 2, which stands for 2 tens, or 20. Multiply the 8 in the tens place by 4, giving 32, then add the carried 2, giving 34. (We actually multiplied 80 by 4, giving 320, and carried 20, which we then added to 320.) Write the 4 in the tens place just to the left of the 4 in the units place, and carry the 3 (which stands for 3 hundreds), placing it over the hundreds column. To finish, multiply the 3 in the hundreds place by 4, giving 12 (actually 1,200), and add the 3 that we carried, giving 15 (actually 1,500): We can follow a similar procedure when both numbers to be multiplied have more than one digit. To multiply 36 by 52, for example, begin by multiplying the top number, 36, by the unit 2 of the bottom number: Next multiply the 6 by 5, giving 30, and put the 0 under the number 7 in the tens place of the partial product. This placement is chosen because the 5 in the bottom number is in the tens place and actually represents 50. Carry the number 3 as usual. Multiply the 3 in the tens place of the top number by 5, giving 15, and add the carried 3, giving 18 (really 5 times 30, plus 30, for a total of 180). Now write the 8 in the hundreds place (directly to the left of the 0 in the tens place), and carry the 1 into the thousands place. We obtain the total product by adding the two partial products: For each digit in the bottom number, there will be a line for a partial product obtained as we multiply out the units, tens, hundreds, and so on. Place each successive partial product one digit to the left of the partial product above it, corresponding to the position of the digit being multiplied in the bottom number. Thus, in the above example, the partial product of 5 times 36 yields 180, and the 0 in the digits place must be written in the tens place of the partial product. This placement results because the 5 in the tens place is really 50 and therefore the entire partial product is really 1,800: D Dividing Positive Integers The arithmetic operation of division is the opposite, or inverse, of multiplication. Using the example of 12 divided by 4, we may indicate division by the division sign (12 ÷ 4), a bar (? a slash (12/4), or the notation p . Division determines how many times one number is contained in another number. For example, 4 is contained 3 times ), in 12; thus, 12 apples could be divided into 3 sets of 4 apples, so 12 divided by 4 is 3. The number to be divided is called the dividend, the number the dividend is divided by is called the divisor, and the end result of division is called the quotient: Simple divisions such as 12 ÷ 4 may be carried out mentally, but more complicated cases require a procedure known as long division. Long division involves the repetition of simple operations. For example, to divide 4,518 by 6, consider the divisor (6) and the first digit (4) of the dividend to see whether the divisor is contained in that first digit one or more times: If the first digit is too small (6 is not contained in 4 even once), try to divide the first two digits of the dividend (45) by the divisor (6) . To determine how many 6s are contained in 45, make a guess. If we guess 8, we can check our guess by multiplying 6 by 8, which yields 48. Since 48 is more than 45, the guess was too big. Guessing 6 and multiplying 6 × 6 yields 36--too small. We know it is too small because when we subtract 36 from 45 (45 - 36) we get 9, which indicates that 45 contains another 6. Therefore 6 will go into 45 no more than 7 times (6 × 7 = 42). Write the number 7 in the quotient over the 5 in the dividend, 4,518, and write the 42 (the product of the divisor, 6, and the first number of the quotient, 7) under the 45 and subtract from it, yielding 3. What we have actually determined so far is that 4,518 contains at least 700 6s, and that 318 is left when these are taken away. In the next step of the division process, bring the 1 in the dividend down and write it to the right of the 3 to give 31: The 6 in the divisor will go into 31 no more than 5 times (6 × 5 = 30). Write the 5 in the quotient to the right of the 6, above the 1 in the dividend. Place the product of 5 × 6, or 30, under the 31 and subtract, yielding 1. Bring the 8 from the dividend down and write it to the right of the 1 to give 18. The 6 in the divisor will go into 18 exactly 3 times, so write the number 3 in the quotient above the 8 in the dividend: The answer to how many times 6 will divide 4,518 is therefore 753. We can verify this solution by multiplying 6 × 753, which yields the dividend 4,518. Repetitious operations such as those employed in long division are called iterative processes and are often used in computer science (see Algorithm). D1 Remainders Dividends are not always evenly divisible by divisors. Had the dividend in the last example been 4,523 rather than 4,518, the final subtraction would not have yielded 0, but would have left a remainder of 5--too small for 6 to go into. We can incorporate this remainder into the answer as a fraction of the divisor, in this case 6, leaving a fractional remainder of ? : D2 Multidigit Divisors We can also use long division if a divisor has more than one digit. The method is similar to the shorter method described above. For example, if we divide 3,626 by 25, 25 will not go into 3 at all, and will go into 36 only once; therefore, we place a 1 over the 6 in 36. Multiply the 25 by 1, write the product (25) under the 36, and subtract from it, yielding 11. Then bring the 2 down to give 112: Because 25 will go into 112 at most 4 times (4 x 25 = 100), write a 4 next to the 1 in the quotient, over the 2 in the dividend. Multiply the divisor by 4, write the product (100) under the 112, and subtract from it, leaving 12. Bring the 6 down for a total of 126. Because 25 will go into 126 five times, with a remainder of 1, the answer is III OPERATIONS WITH NEGATIVE INTEGERS We can most easily visualize negative numbers by considering the familiar numbers of arithmetic, the positive integers, arranged in a line and increasing in a positive direction. Negative numbers are reached by proceeding away from 0 in the opposite direction. The following number line represents both positive and negative numbers: To subtract 4 from 2, count backward four units from 2 to reach the difference: -2. (If no sign appears before a number, we assume the number is positive.) To handle arithmetic operations involving negative numbers, it is necessary to understand the concept of absolute value. A number's absolute value is its magnitude regardless of its sign. Thus, the absolute value of +5 is 5, and the absolute value of -5 is also 5. In symbolic notation, the absolute value of any number a is written as a. The absolute value of 7, for example, is written 7 and is equal to 7, while the absolute value of -9 is written -9 and is equal to 9. A Adding Negative Integers By referring once again to the number line, which includes both positive and negative numbers, we can see that the sum of any two negative numbers is negative and equals the total of the absolute values of the two numbers, but negative: If, however, the signs of the two numbers are opposite, then we must proceed more cautiously. To add -4 and 2, for example, begin at -4 and add 2 by moving 2 units forward, in the positive direction to the right. The result is -2. The answer is the difference between the absolute values of the two numbers, with the result taking the sign of the larger number. The answer to this problem must be negative because in terms of absolute value, -4 > 2. (The symbol > means "greater than.") We obtain the answer, however, by calculating the difference between the absolute values 4 and 2. Similarly, to add -3 and 7, begin at -3 and move 7 units forward, reaching 4. Two rules simplify this process: Rule 1: To add two numbers with the same sign, add the absolute values of the two numbers, then give the answer the sign of the numbers in question: Rule 2: To add two numbers with different signs, determine their absolute values and subtract the smaller number from the larger. Give the result the sign of the number with the larger absolute value: B Subtracting Negative Integers The subtraction of negative numbers is easy to visualize on the number line. Subtracting one number from another involves moving along the number line in the opposite direction from addition. To subtract 5 from 7, for example, begin at 7 and move in the negative direction 5 units, reaching 2 as the answer. Similarly, to subtract -2 from 5, start at 5 but reverse direction and move 2 units in the positive direction to 7. Subtracting -2 basically means adding 2, or more generally, subtracting a negative number is the same thing as adding the absolute value of the number. We can therefore change two negative (or minus) signs in a row into a single positive (or plus) sign. For example: We can change a negative and a positive sign together into a negative sign: C Multiplying Negative Integers Determining the sign of a product is straightforward. To multiply two numbers with the same sign, multiply their absolute values and give the resulting product a positive sign: To multiply two numbers with different signs, multiply their absolute values and give the resulting product a negative sign: D Dividing Negative Integers Division is the inverse of multiplication; therefore, dividing positive or negative numbers involves rules similar to those for multiplication. To divide two numbers with the same sign, divide their absolute values and give the resulting quotient a positive sign: To divide two numbers with different signs, divide their absolute values and give the resulting quotient a negative sign: IV OPERATIONS WITH FRACTIONS Numbers that represent parts of a whole are called fractions or rational numbers. Simple fractions are familiar: a dime is s of a dollar; ?of a pie plus y of a pie is ? of a pie; and so on. In general, we can express fractions as the quotient of two integers a and b : The top number in a fraction is called the numerator and the bottom number is called the denominator. Two types of fractions exist: proper and improper. A proper fraction is one in which the numerator is smaller than the denominator; ? , -o and o are all proper fractions. An improper fraction is one in which the numerator is larger , than the denominator; ? , -? and ?are improper fractions. We can convert improper fractions to mixed fractions or whole numbers (for example, ? = 1y, -?= -2, and ?= , 2 ? by dividing the numerator by the denominator and expressing any remainder as a fraction of the denominator. ) A fraction is said to be reduced to lowest terms if neither the numerator nor the denominator has a factor in common. A factor is a number by which another number can be divided evenly. For example, ?is not reduced to lowest terms because both 6 and 8 have 2 as a factor: Since a number divided by itself is always equal to 1, = 1. Multiplying any number by 1 does not change the number, so × ? = ? . Reduced to lowest terms, then, ?is ? . A Adding and Subtracting Fractions To add or subtract fractions that have the same denominators, add or subtract the numerators according to the rules for integers, and express the result as a fraction of the denominator. The result is normally reduced to lowest terms. For example, Only fractions with equal denominators may be added or subtracted as they stand. If the denominators of fractions to be added are unequal, we must find a common denominator. In the expression ? + ? , for example, the denominators 3 and 4 are different. One quick way to obtain a common denominator for two fractions is to multiply their denominators. In this case that multiplication gives us 12. Thus, to add ? and ? , we should change the fractions into their equivalents with 12 as a common denominator. To do so, multiply both the numerator and denominator of each fraction by the denominator of the other fraction: Multiplying both the numerator and denominator by the same number does not change the overall value of a fraction. Now we can add Subtracting fractions requires the same procedure: B Multiplying Fractions Multiplying two fractions, ¡ and ¢, is straightforward. Simply multiply numerators together and multiply denominators together: For example, The answer has been reduced to its lowest term, in this case ? . The rules for multiplying signed (positive or negative) fractions are the same as those for multiplying signed integers. The same is true for the rules governing addition, subtraction, and division of signed fractions: C Dividing Fractions The division of fractions is most easily understood in terms of reciprocals. Every number (except 0) has a reciprocal, or another number such that the product of the number and its reciprocal equals one. The reciprocal of 3, for example, is ? Zero (0) has no reciprocal, because no number can be multiplied by it to equal 1. Any . number multiplied by zero equals zero. To divide ¡ by ¢, multiply ¡ by the reciprocal of ¢: Division is equivalent to multiplying by the reciprocal--that is, y ÷ ?is the same as y × because both the numerator and the denominator can be multiplied by the same nonzero number without affecting the overall value of the fraction. Multiply both numerator and denominator in the graphic above by ¤: This division is equivalent to multiplying the first number, ¡ , by the reciprocal of the second number--by ¤. The reciprocal of a fraction is simply the fraction flipped upside down. Here is an example that uses actual numbers: Dividing a whole number by a fraction works the same way: V OPERATIONS WITH DECIMALS The idea of place values can be extended to accommodate fractions. Instead of writing 1? (one and two-tenths), we can use a decimal point (.) to represent the same fraction as 1.2. Just as places to the left of the decimal represent units, tens, hundreds, and so on, those to the right of the decimal represent places for tenths (s ), hundredths (t ), thousandths ( ), and so forth. In a decimal number such as 8.632 the numbers to the right of the decimal point represent This number is read "eight and six hundred thirty-two thousandths" or "eight point six three two." Zeroes are often added to the left of the decimal point when a number is less than one. For example, we can write y as either .5 or 0.5. A Adding and Subtracting Decimals Decimals allow us to add and subtract numbers that include fractions just as we add and subtract integers. But we must be careful to always align the decimal point so that tens are under tens, units under units, tenths under tenths, and so on, ensuring that each value is being added to or subtracted from a similar value at every step. For example, to add 365.289 and 32.4, align the decimals and then add the numbers beginning at the right and moving to the left: Extra zeros to the right of a number do not change the number. Filling in the number of places with zeroes can help ensure that the same number of places exists to the right of the decimal point for all numbers being added or subtracted: The decimal point in the sum falls directly beneath the decimal points in the numbers being added. Subtraction with decimals proceeds in much the same way as addition: We can use zeros to ensure that the numbers' place values line up, then subtract the numbers as usual. B Multiplying Decimals Multiplying decimals is similar to multiplying integers, except that the position of the decimal point must be kept in mind. First, multiply decimal numbers as if they were integers, without considering the decimal points. Then place the decimal point at the appropriate position in the product so that the number of decimal places is the same as the total number of decimal places in the numbers being multiplied. For example, in multiplying 0.3 by 0.5 the 15 in the product is the direct value of 3 times 5. We place the decimal point differently than we do in addition and subtraction. Since the factors, 0.3 and 0.5, each have one decimal place, the product must have two decimal places. Thus, 0.15 is the product. Another example will serve to clarify this concept. Multiply 0.2 by 0.3. The product of 2 and 3 is 6, but since 0.2 and 0.3 each have one decimal place, the product must have a total of two decimal places. We can fulfill this requirement by placing the decimal point two places to the left of the 6 in the product, then adding a zero to fill the tenths place: 0.06. More complicated problems are solved similarly: Because the top number has three decimal places and the bottom number has two, the product must have a total of five decimal places. Count five places to the left starting with the digit farthest to the right (the 8 of the final product), adding a zero if necessary. The answer is 0.03108. One final example will also illustrate the importance of counting the proper number of decimal places in determining products of decimal numbers. Multiply .001 and .002. The multiplication of the digits is simple: the answer is 2. However, each number has three decimal places, giving a total of six places that must be preserved in the product. Insert five zeroes to fill the places between the 2 and the decimal point. The answer is 0.000002. C Dividing Decimals Like multiplication, the division of decimal numbers follows the same procedures used to divide integers, except that we must take care to determine the proper placement of decimal points in quotients. Dividing a decimal number by a whole number is straightforward: Place the decimal point in the quotient directly above the decimal point in the dividend and ignore it during the rest of the process of division: In cases where the divisor is a decimal number, convert the problem to one in which the divisor is an integer; division may then proceed as in the above example. To divide 14 by 0.7, for example, convert the divisor to an integer by multiplying it by 10: (0.7)(10) = 7. Then multiply the dividend by an equal amount. We can understand this procedure more easily by considering the division rewritten as a fraction. Multiplying both numerator and denominator by the same amount will not change the value of the fraction: Similarly, the division of 2.675 by 0.23 can be considered in the form 2.675/0.23. We can convert this fraction to a division involving an integer divisor, namely 23, if we multiply both numerator and denominator by 100: We can convert any division problem involving a decimal divisor into a problem with an integer divisor simply by moving the decimal point in the divisor as many places to the right as is necessary to make it an integer. Then move the decimal point in the dividend an equal number of places to the right, and add zeros if necessary. For example, to divide 21.5 by .002, move the decimal point in the divisor three places to the right, giving the integer 2. Move the decimal point in the dividend three places to the right as well: Carry out the division as usual, placing the decimal point in the quotient directly over the new decimal point in the dividend. The quotient in this case is 10750: See also Mathematical Symbols; Mathematics, New; Number. Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.
